Links between cohomology and arithmetic

This thesis consists of three independent chapters. Each chapter deals with a particular link between arithmetic and cohomology. In the first chapter, written together with prof. dr. S.J. Edixhoven, we consider smooth and proper Deligne-Mumford stacks whose number of points over a finite field is a polynomial. The main result is that the cohomology of such stacks, both etale and Betti, is of Tate type. The second chapter generalizes the p-adic De Rham comparison theorem from schemes to Deligne-Mumford stacks. The last chapter deals with Kedlaya's algorithm for counting points of hyperelliptic curves over finite fields. A different basis than the one described in the original algorithm is described, which has the advantage that it is denominator free.